Question

Compute the realized average life of a 30 yr., 5/1 ARM with initial rate of 4%, and a margin of 2%. The underlying index is (date in yrs., index in %): (0,2), (1,3), (2,3), (3,2.5), (4,3), (5,5), (6,5), and (7-30,6). The loan is never prepaid, nor curtailed, nor defaults. There are no rate or payment caps. Annotate excel, or show all work.

Answer 1

A 5/1 ARM 30 mortgage will have interest rate fixed for first 5 years and then change linked to index each year . Hence in this case the 30 year mortgage : First 5 year rate = 4% and 6 year rate = 7% and 7 to 30 year rate = 8%

We will calculate the actual monthly payments and then then weighted monthly payments (weighted by the month paid in ) and then divide to obtain the realised average life.

Monthly payment formula = Loan Amount * [ r * (1+r)^t ] / [ (1+r)^t - 1] ; where r is the monthly interest rate and t is the loan tenure in months. We assume loan amount of $1000

Monthly payment Year 1 to 5 (Month 1 to 60) = 1000 * [(4%/12) * (1+4%/12)³⁶⁰] / [ (1+4/12%)³⁶⁰ - 1] = 4.77

Loan balance at end of month k= Loan Amount * [ (1+r)^t - (1+r)^k] / [ (1+r)^t - 1]; where k is the month at which the loan balance is being calculated

Loan Balance at end of Year 5 (month 60) = 1000 * [(1+ 4%/12)³⁶⁰ - (1+4%/12)⁶⁰] / [ (1+4/12%)³⁶⁰ - 1] = 904.48

Monthly payment Year 6 (residual tenure is 25 years) = 904.48 * [(7%/12) * (1+7%/12)³⁰⁰] / [ (1+4/12%)³⁰⁰ - 1] = 6.39

Loan Balance at end of Year 6 (month 72) = 904.48 * [(1+ 7%/12)³⁰⁰ - (1+7%/12)¹²] / [ (1+7/12%)³⁰⁰ - 1] = 890.64

Monthy payment Year 7 to 30 (residuall tenure 24 years) = 890.64 * [(8%/12) * (1+8%/12)²⁸⁸] / [ (1+8/12%)²⁸⁸ - 1] = 6.97

So now we can put this in the excel sheet:

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v

v

v

Hence we see that he realised average life for this loan is 189.19 months